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Difference between revisions of "Riesz representation theorem"

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(Added the characterization of the dual of C(X,B) when B is finite dimensional.)
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{{MSC|28A33}}
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{{MSC|28A33|28A25}}
  
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
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A central theorem in classical measure theory, sometimes called
 
A central theorem in classical measure theory, sometimes called
 
Riesz-Markov theorem, which states the following.
 
Riesz-Markov theorem, which states the following.
Let $X$ be a compact Hausdorff topological space, $C(X)$ the [[Banach space]]
+
Let $X$ be a compact [[Hausdorff space|Hausdorff]] topological space, $C(X)$ the [[Banach space]]
 
of real valued continuous functions on $X$ and $L: C(X)\to \mathbb R$ a
 
of real valued continuous functions on $X$ and $L: C(X)\to \mathbb R$ a
 
continuous linear functional which is nonnegative, i.e. such that $L(f)\geq 0$
 
continuous linear functional which is nonnegative, i.e. such that $L(f)\geq 0$
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L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, .
 
L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, .
 
\]
 
\]
 +
The theorem is often stated in a more general version for ''locally compact'' Hausdorff spaces $X$, of which the statement above is a simple corollary (cp, with Section 2.14 of {{Cite|Ru}})
  
 
An analogous statement which is commonly referred to as Riesz representation theorem
 
An analogous statement which is commonly referred to as Riesz representation theorem
 
is that, under the assumptions above, the dual of $C(X)$ is the space $\mathcal{M}^b (X)$
 
is that, under the assumptions above, the dual of $C(X)$ is the space $\mathcal{M}^b (X)$
of  $\mathbb R$-valued measures with finite total variation (cp. with [[Convergence of measures]] for the relevant definitions). Combined with  the [[Radon-Nikodým theorem]],
+
of  $\mathbb R$-valued measures with finite total variation (cp. with [[Signed measure]] for the relevant definitions). Combined with  the [[Radon-Nikodým theorem]],
 
this amounts to the following alternative statement: for any element $L\in (C(X))'$
 
this amounts to the following alternative statement: for any element $L\in (C(X))'$
 
there are a Radon measure $\mu$ and a Borel function $g$
 
there are a Radon measure $\mu$ and a Borel function $g$
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L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, .
 
L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, .
 
\]
 
\]
 
More general statements for locally compact Hausdorff spaces can
 
be easily derived from the ones above.
 
  
 
The statement can be also generalized to a similar description of the dual of $C(X,B)$ when $B$ is Banach space. For instance, if $B$ is a finite-dimensional space, then for any $L\in C(X,B)'$ there is a Radon measure $\mu$ on $X$ and a Borel measurable map $g: X\to B'$ such that $\|g\|_{B'}=1$ $\mu$-a.e. and
 
The statement can be also generalized to a similar description of the dual of $C(X,B)$ when $B$ is Banach space. For instance, if $B$ is a finite-dimensional space, then for any $L\in C(X,B)'$ there is a Radon measure $\mu$ on $X$ and a Borel measurable map $g: X\to B'$ such that $\|g\|_{B'}=1$ $\mu$-a.e. and
 
\[
 
\[
L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X)\, .
+
L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X, B)\, .
 
\]
 
\]
  
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|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
 
|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
 
|-
 
|-
|valign="top"|{{Ref|Ma}}||  P. Mattila, "Geometry of sets and measures in euclidean spaces.  Cambridge Studies in Advanced Mathematics, 44. Cambridge University  Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
+
|valign="top"|{{Ref|Ma}}||  P. Mattila, "Geometry of sets and measures in euclidean spaces".  Cambridge Studies in Advanced Mathematics, 44. Cambridge University  Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
 +
|-
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|valign="top"|{{Ref|Ru}}|| W. Rudin, "Real and complex analysis". McGraw-Hill Book Co., New York-Toronto, Ont.-London  1966 {{MR|021052}} {{ZBL|0142.01701}}
 
|-
 
|-
 
|}
 
|}

Latest revision as of 17:37, 18 August 2012

2010 Mathematics Subject Classification: Primary: 28A33 Secondary: 28A25 [MSN][ZBL]

A central theorem in classical measure theory, sometimes called Riesz-Markov theorem, which states the following. Let $X$ be a compact Hausdorff topological space, $C(X)$ the Banach space of real valued continuous functions on $X$ and $L: C(X)\to \mathbb R$ a continuous linear functional which is nonnegative, i.e. such that $L(f)\geq 0$ whenever $f\geq 0$. Then there is a Radon measure $\mu$ on the $\sigma$-algebra of Borel sets $\mathcal{B} (X)$ such that \[ L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, . \] The theorem is often stated in a more general version for locally compact Hausdorff spaces $X$, of which the statement above is a simple corollary (cp, with Section 2.14 of [Ru])

An analogous statement which is commonly referred to as Riesz representation theorem is that, under the assumptions above, the dual of $C(X)$ is the space $\mathcal{M}^b (X)$ of $\mathbb R$-valued measures with finite total variation (cp. with Signed measure for the relevant definitions). Combined with the Radon-Nikodým theorem, this amounts to the following alternative statement: for any element $L\in (C(X))'$ there are a Radon measure $\mu$ and a Borel function $g$ such that $|g|=1$ $\mu$-a.e. and \[ L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, . \]

The statement can be also generalized to a similar description of the dual of $C(X,B)$ when $B$ is Banach space. For instance, if $B$ is a finite-dimensional space, then for any $L\in C(X,B)'$ there is a Radon measure $\mu$ on $X$ and a Borel measurable map $g: X\to B'$ such that $\|g\|_{B'}=1$ $\mu$-a.e. and \[ L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X, B)\, . \]

References

[AmFuPa] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ru] W. Rudin, "Real and complex analysis". McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966 MR021052 Zbl 0142.01701
How to Cite This Entry:
Riesz representation theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_representation_theorem&oldid=27173